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Theorem List for Metamath Proof Explorer - 30901-31000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcdlemd6 30901 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 31-May-2012.)

Theoremcdlemd7 30902 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Jun-2012.)

Theoremcdlemd8 30903 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Jun-2012.)

Theoremcdlemd9 30904 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 2-Jun-2012.)

Theoremcdlemd 30905 If two translations agree at any atom not under the fiducial co-atom , then they are equal. Lemma D in [Crawley] p. 113. (Contributed by NM, 2-Jun-2012.)

Theoremltrneq3 30906 Two translations agree at any atom not under the fiducial co-atom iff they are equal. (Contributed by NM, 25-Jul-2013.)

Theoremcdleme00a 30907 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.)

Theoremcdleme0aa 30908 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.)

Theoremcdleme0a 30909 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 12-Jun-2012.)

Theoremcdleme0b 30910 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.)

Theoremcdleme0c 30911 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 12-Jun-2012.)

Theoremcdleme0cp 30912 Part of proof of Lemma E in [Crawley] p. 113. TODO: Reformat as in cdlemg3a 31295- swap consequent equality; make antecedent use df-3an 938. (Contributed by NM, 13-Jun-2012.)

Theoremcdleme0cq 30913 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-Apr-2013.)

Theoremcdleme0dN 30914 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.)

Theoremcdleme0e 30915 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.)

Theoremcdleme0fN 30916 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.)

Theoremcdleme0gN 30917 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.)

Theoremcdlemeulpq 30918 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 5-Dec-2012.)

Theoremcdleme01N 30919 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)

Theoremcdleme02N 30920 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)

Theoremcdleme0ex1N 30921* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)

Theoremcdleme0ex2N 30922* Part of proof of Lemma E in [Crawley] p. 113. Note that is a shorter way to express . (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)

Theoremcdleme0moN 30923* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)

Theoremcdleme1b 30924 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma showing is a lattice element. represents their f(r). (Contributed by NM, 6-Jun-2012.)

Theoremcdleme1 30925 Part of proof of Lemma E in [Crawley] p. 113. represents their f(r). Here we show r f(r) = r u (7th through 5th lines from bottom on p. 113). (Contributed by NM, 4-Jun-2012.)

Theoremcdleme2 30926 Part of proof of Lemma E in [Crawley] p. 113. . represents f(r). is the fiducial co-atom (hyperplane) w. Here we show that (r f(r)) w = u in their notation (4th line from bottom on p. 113). (Contributed by NM, 5-Jun-2012.)

Theoremcdleme3b 30927 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 30934 and cdleme3 30935. (Contributed by NM, 6-Jun-2012.)

Theoremcdleme3c 30928 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 30934 and cdleme3 30935. (Contributed by NM, 6-Jun-2012.)

Theoremcdleme3d 30929 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 30934 and cdleme3 30935. (Contributed by NM, 6-Jun-2012.)

Theoremcdleme3e 30930 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 30934 and cdleme3 30935. (Contributed by NM, 6-Jun-2012.)

Theoremcdleme3fN 30931 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 30934 and cdleme3 30935. TODO: Delete - duplicates cdleme0e 30915. (Contributed by NM, 6-Jun-2012.) (New usage is discouraged.)

Theoremcdleme3g 30932 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 30934 and cdleme3 30935. (Contributed by NM, 7-Jun-2012.)

Theoremcdleme3h 30933 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 30934 and cdleme3 30935. (Contributed by NM, 6-Jun-2012.)

Theoremcdleme3fa 30934 Part of proof of Lemma E in [Crawley] p. 113. See cdleme3 30935. (Contributed by NM, 6-Oct-2012.)

Theoremcdleme3 30935 Part of proof of Lemma E in [Crawley] p. 113. represents f(r). is the fiducial co-atom (hyperplane) w. Here and in cdleme3fa 30934 above, we show that f(r) W (4th line from bottom on p. 113), meaning it is an atom and not under w, which in our notation is expressed as . Their proof provides no details of our lemmas cdleme3b 30927 through cdleme3 30935, so there may be a simpler proof that we have overlooked. (Contributed by NM, 7-Jun-2012.)

Theoremcdleme4 30936 Part of proof of Lemma E in [Crawley] p. 113. and represent f(s) and fs(r). Here show p q = r u at the top of p. 114. (Contributed by NM, 7-Jun-2012.)

Theoremcdleme4a 30937 Part of proof of Lemma E in [Crawley] p. 114 top. represents fs(r). Auxiliary lemma derived from cdleme5 30938. We show fs(r) p q. (Contributed by NM, 10-Nov-2012.)

Theoremcdleme5 30938 Part of proof of Lemma E in [Crawley] p. 113. represents fs(r). We show r fs(r)) = p q at the top of p. 114. (Contributed by NM, 7-Jun-2012.)

Theoremcdleme6 30939 Part of proof of Lemma E in [Crawley] p. 113. This expresses (r fs(r)) w = u at the top of p. 114. (Contributed by NM, 7-Jun-2012.)

Theoremcdleme7aa 30940 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 30946 and cdleme7 30947. (Contributed by NM, 7-Jun-2012.)

Theoremcdleme7a 30941 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 30946 and cdleme7 30947. (Contributed by NM, 7-Jun-2012.)

Theoremcdleme7b 30942 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 30946 and cdleme7 30947. (Contributed by NM, 7-Jun-2012.)

Theoremcdleme7c 30943 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 30946 and cdleme7 30947. (Contributed by NM, 7-Jun-2012.)

Theoremcdleme7d 30944 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 30946 and cdleme7 30947. (Contributed by NM, 8-Jun-2012.)

Theoremcdleme7e 30945 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 30946 and cdleme7 30947. (Contributed by NM, 8-Jun-2012.)

Theoremcdleme7ga 30946 Part of proof of Lemma E in [Crawley] p. 113. See cdleme7 30947. (Contributed by NM, 8-Jun-2012.)

Theoremcdleme7 30947 Part of proof of Lemma E in [Crawley] p. 113. and represent fs(r) and f(s) respectively. is the fiducial co-atom (hyperplane) that they call w. Here and in cdleme7ga 30946 above, we show that fs(r) W (top of p. 114), meaning it is an atom and not under w, which in our notation is expressed as . (Note that we do not have a symbol for their W.) Their proof provides no details of our cdleme7aa 30940 through cdleme7 30947, so there may be a simpler proof that we have overlooked. (Contributed by NM, 9-Jun-2012.)

Theoremcdleme8 30948 Part of proof of Lemma E in [Crawley] p. 113, 2nd paragraph on p. 114. represents s1. In their notation, we prove p s1 = p s. (Contributed by NM, 9-Jun-2012.)

Theoremcdleme9a 30949 Part of proof of Lemma E in [Crawley] p. 113. represents s1, which we prove is an atom. (Contributed by NM, 10-Jun-2012.)

Theoremcdleme9b 30950 Utility lemma for Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Oct-2012.)

Theoremcdleme9 30951 Part of proof of Lemma E in [Crawley] p. 113, 2nd paragraph on p. 114. and represent s1 and f(s) respectively. In their notation, we prove f(s) s1 = q s1. (Contributed by NM, 10-Jun-2012.)

Theoremcdleme10 30952 Part of proof of Lemma E in [Crawley] p. 113, 2nd paragraph on p. 114. represents s2. In their notation, we prove s s2 = s r. (Contributed by NM, 9-Jun-2012.)

Theoremcdleme8tN 30953 Part of proof of Lemma E in [Crawley] p. 113, 2nd paragraph on p. 114. represents t1. In their notation, we prove p t1 = p t. (Contributed by NM, 8-Oct-2012.) (New usage is discouraged.)

Theoremcdleme9taN 30954 Part of proof of Lemma E in [Crawley] p. 113. represents t1, which we prove is an atom. (Contributed by NM, 8-Oct-2012.) (New usage is discouraged.)

Theoremcdleme9tN 30955 Part of proof of Lemma E in [Crawley] p. 113, 2nd paragraph on p. 114. and represent t1 and f(t) respectively. In their notation, we prove f(t) t1 = q t1. (Contributed by NM, 8-Oct-2012.) (New usage is discouraged.)

Theoremcdleme10tN 30956 Part of proof of Lemma E in [Crawley] p. 113, 2nd paragraph on p. 114. represents t2. In their notation, we prove t t2 = t r. (Contributed by NM, 8-Oct-2012.) (New usage is discouraged.)

Theoremcdleme16aN 30957 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, showing, in their notation, s u t u. (Contributed by NM, 9-Oct-2012.) (New usage is discouraged.)

Theoremcdleme11a 30958 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 30968. (Contributed by NM, 12-Jun-2012.)

Theoremcdleme11c 30959 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 30968. (Contributed by NM, 13-Jun-2012.)

Theoremcdleme11dN 30960 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 30968. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.)

Theoremcdleme11e 30961 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 30968. (Contributed by NM, 13-Jun-2012.)

Theoremcdleme11fN 30962 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 30968. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.)

Theoremcdleme11g 30963 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 30968. (Contributed by NM, 14-Jun-2012.)

Theoremcdleme11h 30964 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 30968. (Contributed by NM, 14-Jun-2012.)

Theoremcdleme11j 30965 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 30968. (Contributed by NM, 14-Jun-2012.)

Theoremcdleme11k 30966 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 30968. (Contributed by NM, 15-Jun-2012.)

Theoremcdleme11l 30967 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 30968. (Contributed by NM, 15-Jun-2012.)

Theoremcdleme11 30968 Part of proof of Lemma E in [Crawley] p. 113, 1st sentence of 3rd paragraph on p. 114. and represent f(s) and f(t) respectively. Their proof provides no details of our cdleme11a 30958 through cdleme11 30968, so there may be a simpler proof that we have overlooked. (Contributed by NM, 15-Jun-2012.)

Theoremcdleme12 30969 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, first part of 3rd sentence. and represent f(s) and f(t) respectively. (Contributed by NM, 16-Jun-2012.)

Theoremcdleme13 30970 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, "<s,t,p> and <f(s),f(t),q> are centrally perspective." and represent f(s) and f(t) respectively. (Contributed by NM, 7-Oct-2012.)

Theoremcdleme14 30971 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, "<s,t,p> and <f(s),f(t),q> ... are axially perspective." We apply dalaw 30584 to cdleme13 30970. and represent f(s) and f(t) respectively. (Contributed by NM, 8-Oct-2012.)

Theoremcdleme15a 30972 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, showing, in their notation, ((s p) (f(s) q)) ((t p) (f(t) q))=((p s1) (q s1)) ((p t1) (q t1)). We represent f(s), f(t), s1, and t1 with , , , and respectively. The order of our operations is slightly different. (Contributed by NM, 9-Oct-2012.)

Theoremcdleme15b 30973 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, showing, in their notation, (p s1) (q s1)=s1. We represent s1 with . (Contributed by NM, 10-Oct-2012.)

Theoremcdleme15c 30974 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, showing, in their notation, ((p s1) (q s1)) ((p t1) (q t1))=s1 t1. and represent s1 and t1 respectively. The order of our operations is slightly different. (Contributed by NM, 10-Oct-2012.)

Theoremcdleme15d 30975 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, showing, in their notation, s1 t1 w. and represent s1 and t1 respectively. The order of our operations is slightly different. (Contributed by NM, 10-Oct-2012.)

Theoremcdleme15 30976 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, showing, in their notation, (s t) (f(s) f(t)) w. We use , for f(s), f(t) respectively. (Contributed by NM, 10-Oct-2012.)

Theoremcdleme16b 30977 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, first part of 3rd sentence. and represent f(s) and f(t) respectively. It is unclear how this follows from s u t u, as the authors state, and we used a different proof. (Note: the antecedent is not used.) (Contributed by NM, 11-Oct-2012.)

Theoremcdleme16c 30978 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, 2nd part of 3rd sentence. and represent f(s) and f(t) respectively. We show, in their notation, s t f(s) f(t)=s t u. (Contributed by NM, 11-Oct-2012.)

Theoremcdleme16d 30979 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, 3rd part of 3rd sentence. and represent f(s) and f(t) respectively. We show, in their notation, (s t) (f(s) f(t)) is an atom. (Contributed by NM, 11-Oct-2012.)

Theoremcdleme16e 30980 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, 3rd part of 3rd sentence. and represent f(s) and f(t) respectively. We show, in their notation, (s t) (f(s) f(t))=(s t) w. (Contributed by NM, 11-Oct-2012.)

Theoremcdleme16f 30981 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, 3rd part of 3rd sentence. and represent f(s) and f(t) respectively. We show, in their notation, (s t) (f(s) f(t))=(f(s) f(t)) w. (Contributed by NM, 11-Oct-2012.)

Theoremcdleme16g 30982 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, Eq. (1). and represent f(s) and f(t) respectively. We show, in their notation, (s t) w=(f(s) f(t)) w. (Contributed by NM, 11-Oct-2012.)

Theoremcdleme16 30983 Part of proof of Lemma E in [Crawley] p. 113, conclusion of 3rd paragraph on p. 114. and represent f(s) and f(t) respectively. We show, in their notation, (s t) w=(f(s) f(t)) w, whether or not u s t. (Contributed by NM, 11-Oct-2012.)

Theoremcdleme17a 30984 Part of proof of Lemma E in [Crawley] p. 114, first part of 4th paragraph. , , and represent f(s), fs(p), and s1 respectively. We show, in their notation, fs(p)=(p q) (q s1). (Contributed by NM, 11-Oct-2012.)

Theoremcdleme17b 30985 Lemma leading to cdleme17c 30986. (Contributed by NM, 11-Oct-2012.)

Theoremcdleme17c 30986 Part of proof of Lemma E in [Crawley] p. 114, first part of 4th paragraph. represents s1. We show, in their notation, (p q) (q s1)=q. (Contributed by NM, 11-Oct-2012.)

Theoremcdleme17d1 30987 Part of proof of Lemma E in [Crawley] p. 114, first part of 4th paragraph. , represent f(s), fs(p) respectively. We show, in their notation, fs(p)=q. (Contributed by NM, 11-Oct-2012.)

Theoremcdleme0nex 30988* Part of proof of Lemma E in [Crawley] p. 114, 4th line of 4th paragraph. Whenever (in their terminology) p q/0 (i.e. the sublattice from 0 to p q) contains precisely three atoms, any atom not under w must equal either p or q. (In case of 3 atoms, one of them must be u - see cdleme0a 30909- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 30042, our is a shorter way to express . Thus, the negated existential condition states there are no atoms different from p or q that are also not under w. (Contributed by NM, 12-Nov-2012.)

Theoremcdleme18a 30989 Part of proof of Lemma E in [Crawley] p. 114, 2nd sentence of 4th paragraph. , represent f(s), fs(q) respectively. We show fs(q) w. (Contributed by NM, 12-Oct-2012.)

Theoremcdleme18b 30990 Part of proof of Lemma E in [Crawley] p. 114, 2nd sentence of 4th paragraph. , represent f(s), fs(q) respectively. We show fs(q) q. (Contributed by NM, 12-Oct-2012.)

Theoremcdleme18c 30991* Part of proof of Lemma E in [Crawley] p. 114, 2nd sentence of 4th paragraph. , represent f(s), fs(q) respectively. We show fs(q) = p whenever p q has three atoms under it (implied by the negated existential condition). (Contributed by NM, 10-Nov-2012.)

Theoremcdleme22gb 30992 Utility lemma for Lemma E in [Crawley] p. 115. (Contributed by NM, 5-Dec-2012.)

Theoremcdleme18d 30993* Part of proof of Lemma E in [Crawley] p. 114, 4th sentence of 4th paragraph. , , , represent f(s), fs(r), f(t), ft(r) respectively. We show fs(r)=ft(r) for all possible r (which must equal p or q in the case of exactly 3 atoms in p q/0 i.e. when ...). (Contributed by NM, 12-Nov-2012.)

Theoremcdlemesner 30994 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. (Contributed by NM, 13-Nov-2012.)

Theoremcdlemedb 30995 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. represents s2. (Contributed by NM, 20-Nov-2012.)

Theoremcdlemeda 30996 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. represents s2. (Contributed by NM, 13-Nov-2012.)

Theoremcdlemednpq 30997 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. represents s2. (Contributed by NM, 18-Nov-2012.)

TheoremcdlemednuN 30998 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. represents s2. (Contributed by NM, 18-Nov-2012.) (New usage is discouraged.)

Theoremcdleme20zN 30999 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. (Contributed by NM, 17-Nov-2012.) (New usage is discouraged.)

Theoremcdleme20y 31000 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. (Contributed by NM, 17-Nov-2012.)

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